By [1]}, a left cancellative left semi-braces \((B,+,\circ )\)  gives a left non-degenerate solution by \(\lambda _a(b):=a\circ (a^-+b)\)  and \(\rho _b(a):=(a^-+b)^-\circ b\)  for all \(a,b\in B\) . On the other hand, left braces and their generalizations have many connections with other topics such as Hopf-Galois extensions, regular subgroups, rings and near rings, left-orderable groups: for this reason, they increased of interest indipendently of the Yang-Baxter equation.
At first, several authors classified large classes of left braces and investigated the structure of left braces having specific properties.
In [2]}, [3]} Rump completely classified the left braces for which \((B,+)\)  is a cyclic group, while in
 he developed a machinery to construct finite left braces \((B,+,\circ )\)  having fixed group \((B,\circ )\) . In  Smocktunowicz also focused on the group \((B,\circ )\) : indeed, she investigated left braces \((B,+,\circ )\)  for which \((B,\circ )\)  is a nilpotent group. Other authors classified left braces having specific size: in [4]} Bachiller classified left braces of size \(p^3\)  and in [5]} Dietzel classified the ones having size \(p^2 q\) , with \(p,q\)  prime numbers and \(q>p+1\) . Infinite left braces are also considered: for example, in  a classification of left braces \((B,+,\circ )\)  such that \((B,\circ )\)  is an infinite cyclic group was given.
